
\(\Lambda _c\ne 0\)  and \(T=0\) : In this case, we reproduce the results obtained for MAdS background (REF ). The leading contribution comes from the AdS boundary which corresponds to the HMI between our considered subsystems and the non-conformal effects appear as the sub-leading terms. These effects are always negative and therefore the non-conformality decreases \(\hat{\mathcal {I}}(l\Lambda _c ,l^{\prime }\Lambda _c )\) .

\(\Lambda _c=0\)  and \(T\ne 0\) : In this case, we reproduce the previous results obtained for AdS black hole, see [1]}. The leading term corresponds to the zero temperature HMI and the finite temperature correction appears as the sub-leading term. The constant \(\bar{\kappa }_1\)  is positive and therefore, the thermal fluctuations decrease \(\tilde{\mathcal {I}}(lT,l^{\prime }T)\) .

\(\Lambda _c= 0\)  and \(T=0\) : We reach the previous results obtained for pure AdS\(_5\)  which corresponds to the HMI in the zero temperature conformal field theory [1]}. Obviously, this term is positive.

